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Article

Modeling of Multiple Master–Slave Control underIsland Microgrid and Stability Analysis Based onControl Parameter Configuration

Haifeng Liang 1,*, Yue Dong 1, Yuxi Huang 1, Can Zheng 2 and Peng Li 1

1 Department of Electrical Engineering, North China Electric Power University, Baoding 071003, China;[email protected] (Y.D.); [email protected] (Y.H.); [email protected] (P.L.)

2 State Grid Jiaxing Power Supply Company, Jiaxing 314000, China; [email protected]* Correspondence: [email protected]; Tel.: +86-139-3085-6912

Received: 16 July 2018; Accepted: 17 August 2018; Published: 24 August 2018

Abstract: The stable operation of a microgrid is crucial to the integration of renewable energy sources.However, with the expansion of scale in electronic devices applied in the microgrid, the interactionbetween voltage source converters poses a great threat to system stability. In this paper, the modelof a three-source microgrid with a multi master–slave control method in islanded mode is builtfirst of all. Two sources out of three use droop control as the main control source, and another is asubordinate one with constant power control which is also known as real and reactive power (PQ)control. Then, the small signal decoupling control model and its stability discriminant equationare established combined with “virtual impedance”. To delve deeper into the interaction betweenconverters, mutual influence of paralleled converters of two main control micro sources and theireffect on system stability is explored from the perspective of control parameters. Finally, simulationand analysis are launched and the study serves as a reference for parameter setting of converters ina microgrid.

Keywords: islanded mode; multi-source microgrid; stability; small-signal model; multiplemaster–slave control

1. Introduction

Increasing exhaustion of fossil fuel and concerns about environmental pollution have led toextensive exploration of alternative energy sources. Of special interest are renewable energy sources(RES) such as solar and wind energy generation. This has resulted in the emergence of distributedgenerators (DGs) and the concept of microgrid (MG) has been introduced for proper utilization ofDGs [1,2]. As different DGs have different characteristics and natural uncertainty, their presence canhave negative effects on MG. Deviations of voltage and frequency are larger when MG operates inislanded mode in comparison with grid-connected mode [3,4]. Therefore, research on the stability ofMG in islanded mode is of vital importance.

Usually, voltage source converters (VSCs) [5] are used to interconnect different DGs, whoseoutput is mostly in the form of direct or non-common frequency alternating current. Although theincreasing number of power electronic converters improves the control speed, it brings fuzzy influenceon stability owing to the mutual influence between them [6]. To address the abovementioned issue,various methods have been proposed in the literature. In [7,8], the sensitivity of load voltage andvoltage compensation term are introduced to control the bus voltage as well as improve the responsespeed and accuracy. However, limitations in applicable scope of load and voltage reduce its practicality.In [9,10], energy storage devices are used not only to suppress the transient power fluctuations, butalso to greatly improve MGs’ stability thanks to a super-capacitor. However, this method affects

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Energies 2018, 11, 2223 2 of 18

the entire system and declines its inertia. In [11,12], a model that contains converters, controllersand alternating current (AC) grids is built, and high order state space expressions are formulated todetermine its stability. However, the mutual influence between converters is neglected and a largenumber of equations make stability analysis more complex. To improve the accuracy of analysis,the model of MG has been improved for higher accuracy [13–16], including a linear model of MGcombined with a boost converters model [13]; a global model ignoring the change in rotating anglesand coupling terms between DGs [14]; a dynamic model of multi-module comprising multi-classof DGs and loads [15]; and a new type of small-signal model containing secondary control thatanalyzed the stability of the system with eigenvalue [16]. However, the models established in theabove literature didn’t take into account the impact of electronic devices in grids. In view of thisproblem, some research delves deeper into the impact of control parameters on system stability [17].It employs the root locus method to explore the reasonable ranges of control parameters by a smallsignal model. When multi-sources are paralleled, droop control can achieve coordination controlwithout communication and the master–slave control method considers the power tracking of PV andwind power and the system stability [18,19]. However, the influence in different control parameters onthe system is not considered, for their parameters are partly or almost the same.

In view of the problem in [18], this paper builds a model of multi-source MG, emphasizingthe exploration of the mutual influence between converters on stability. To satisfy the decouplingcondition, “virtual impedance” is introduced and a three-source decoupling small-signal model is builtin addition to the characteristic polynomial also being deducted. On this basis, the mutual influencebetween converters on stability is analyzed on the level of control parameters by means of simulationstudies. This method and the result can provide a reference for parameters’ configuration of convertersin MG with a large number of electronic devices plugged in.

2. The Structure of Multi-Source Microgrid and Analysis of Its Problems

The diversity of DGs is a common feature of MGs [15]. Therefore, adopting appropriate controlmethods is an accurate and fast way to achieve the stable operation of islanded MG.

2.1. Multi Master–Slave Control Structure of MG

The traditional master–slave control method consists of one master controller and several slavecontrollers, which forms the energy supply system (ESS), in order to take into account the powertracking the renewable energy and the system stability [19]. With the increase in the permeability andscale of MG, however, a single master controller can no longer maintain the regulation of voltage andfrequency. Hence, several master control sources should be set up with a droop control method. Thissolution is adopted in DGs (e.g., diesel engine, micro gas turbine, fuel cell, etc.), which regulate busvoltage and frequency. Relatively, photovoltaic and wind turbines with PQ control methods run asslave sources. The use of droop control alone implements the peer-to-peer control method, and itrealizes the peer-to-peer control method, which can achieve local control and coordination controlwithout communication.

Figure 1 shows a simplified structure of three-source MG in islanded mode. In Figure 1,Vodn + jVoqn and Rn + jXn are the output voltage of DGn and the equivalent impedance betweenbus and its converter; idn + jiqn is the output current of DGn (n = 1, 2, 3); UGd + jUGq is the bus voltageof MG. DG1 and DG2 serve as master controllers adopting a droop control method, embodying theflexibility and redundancy of a peer-to-peer control method in power vacancy allocation; DG3 servesas a slave controller adopting the PQ control method, ensuring its constant output in islanded mode;Load is comprised of resistance and reactance. This system adopts a master–slave control methodwhen three DGs work and it adopts a peer-to-peer control method when DG3 quits running.

Energies 2018, 11, 2223 3 of 18Energies 2018, 11, x FOR PEER REVIEW 3 of 19

DG1

Load

R1+jX1 R2+jX2 R3+jX3

R4+jX4

UGd+jUGq

Vod1+jVoq1 Vod2+jVoq2 Vod3+jVoq3

id1+jiq1 id2+jiq2 id3+jiq3

id4+jiq4

DG2 DG3

ESS

Figure 1. System model of microgrid (MG) with multiple master–slave control method.

2.2. The Problem under Study

Fully symmetrical structure is mostly used in research on multi-source MG, whose parameters are the same in the same control method, ignoring the independence of control parameters of different DGs. The goal of this paper is to evaluate the following aspects: (1) whether there are interactions between control parameters to maintain stability when several converters operate in parallel; (2) whether there are interactions between control parameters of paralleled master controllers when a system’s control strategy changes from single master–slave control to multiple master–slave control method; and (3) how these interactions influence the stability of MG.

This paper verified the influence of droop coefficient difference on a multi master–slave control method based on the system model as shown in Figure 1. Set up the active droop coefficients of DG1, DG2 as 5 × 10−5, 10 × 10−5; and the reactive droop coefficients of them as 0.003; other parameter settings are shown in Appendix A; the waveform of bus voltage under different working conditions is shown in Figure 2. Figure 2a indicates the public bus bar voltage waveform when it works with DG1 and DG3, which contains only one master micro source; similarly, Figure 2b indicates the bus voltage waveform when it works with DG2 and DG3. The bus voltage can be kept stable in the two cases. Under the same coefficient setting, when DG1, DG2 and DG3 run together, the bus voltage waveform is oscillating and the microgrid is unstable as shown in Figure 2c. The cause of this phenomenon may be that the Cooperative operation of DG1 and DG2 makes the two controllers interact with each other, so that the operation of the microgrid is deviated from its stable running point, and then the instability occurs.

Based on this conjecture, this paper delves deep into the interaction between control parameters under multiple master controllers’ conditions, combining the case with the theoretical model.

Figure 1. System model of microgrid (MG) with multiple master–slave control method.

2.2. The Problem under Study

Fully symmetrical structure is mostly used in research on multi-source MG, whose parameters arethe same in the same control method, ignoring the independence of control parameters of different DGs.The goal of this paper is to evaluate the following aspects: (1) whether there are interactions betweencontrol parameters to maintain stability when several converters operate in parallel; (2) whetherthere are interactions between control parameters of paralleled master controllers when a system’scontrol strategy changes from single master–slave control to multiple master–slave control method;and (3) how these interactions influence the stability of MG.

This paper verified the influence of droop coefficient difference on a multi master–slave controlmethod based on the system model as shown in Figure 1. Set up the active droop coefficients of DG1,DG2 as 5 × 10−5, 10 × 10−5; and the reactive droop coefficients of them as 0.003; other parametersettings are shown in Appendix A; the waveform of bus voltage under different working conditions isshown in Figure 2. Figure 2a indicates the public bus bar voltage waveform when it works with DG1and DG3, which contains only one master micro source; similarly, Figure 2b indicates the bus voltagewaveform when it works with DG2 and DG3. The bus voltage can be kept stable in the two cases.Under the same coefficient setting, when DG1, DG2 and DG3 run together, the bus voltage waveformis oscillating and the microgrid is unstable as shown in Figure 2c. The cause of this phenomenonmay be that the Cooperative operation of DG1 and DG2 makes the two controllers interact with eachother, so that the operation of the microgrid is deviated from its stable running point, and then theinstability occurs.

Based on this conjecture, this paper delves deep into the interaction between control parametersunder multiple master controllers’ conditions, combining the case with the theoretical model.


0 0.1 0.2 0.3 0.4 0.5 0.6Time/s

-800

-600

-400

-200

0

200

400

600

800

(a)

Busb

ar vo

ltage

/V

800

0 0.1 0.2 0.3 0.4 0.5 0.6/s

-800

-600

-400

-200

0

200

400

600

Time

0 0.1 0.2 0.3 0.4 0.5 0.6/s

-1500

-1000

-500

0

500

1000

1500

(c)Time

(b)

Busb

ar vo

ltage

\VBu

sbar

volta

ge\V

Figure 2. Waveforms of bus voltage under different working conditions in case of (a) DG1 and DG3 running; (b) DG2 and DG3 running; (c) DG1, DG2 and DG3 running together.

3. Theoretical Analysis of Control Methods

3.1. PQ Control Method

To improve the response speed and accuracy of power distribution, the PQ control method is adopted [20–22]. Double-loop control structure is employed: the inner loop uses current control to ensure response speed, thereby improving the operation characteristics of the system; the outer loop adopts power control [23].

The structure of inner current loop is shown in Figure 3 [24].

−

sq

sd

VSC control modelInner loop current control

iiip

iiip

*sq

*sd

sd

sq

+

+−

+−

+

+

−− sq

d d

sd

+

+

−

−−

+q q

Figure 3. Structure of inner current loop.

Figure 2. Waveforms of bus voltage under different working conditions in case of (a) DG1 and DG3running; (b) DG2 and DG3 running; (c) DG1, DG2 and DG3 running together.



To improve the response speed and accuracy of power distribution, the PQ control method isadopted [20–22]. Double-loop control structure is employed: the inner loop uses current control toensure response speed, thereby improving the operation characteristics of the system; the outer loopadopts power control [23].


Energies 2018, 11, x FOR PEER REVIEW 4 of 19

0 0.1 0.2 0.3 0.4 0.5 0.6Time/s

-800

-600

-400

-200

0

200

400

600

800

(a)

Busb

ar vo

ltage

/V

800

0 0.1 0.2 0.3 0.4 0.5 0.6/s

-800

-600

-400

-200

0

200

400

600

Time

0 0.1 0.2 0.3 0.4 0.5 0.6/s

-1500

-1000

-500

0

500

1000

1500

(c)Time

(b)

Busb

ar vo

ltage

\VBu

sbar

volta

ge\V

Figure 2. Waveforms of bus voltage under different working conditions in case of (a) DG1 and DG3 running; (b) DG2 and DG3 running; (c) DG1, DG2 and DG3 running together.



To improve the response speed and accuracy of power distribution, the PQ control method is adopted [20–22]. Double-loop control structure is employed: the inner loop uses current control to ensure response speed, thereby improving the operation characteristics of the system; the outer loop adopts power control [23].


−

sq

sd

VSC control modelInner loop current control

iiip

iiip

*sq

*sd

sd

sq

+

+−

+−

+

+

−− sq

d d

sd

+

+

−

−−

+q q

Figure 3. Structure of inner current loop. Figure 3. Structure of inner current loop.

Energies 2018, 11, 2223 5 of 18

In Figure 3, Vcd and Vcq are the d,q axis voltage component of converters and Vod and Voq are thed,q axis voltage components of the filter; Lf is the inductance of filter; Kip and Kii are the proportionaland integral parameters of the proportional-integral (PI) controller.

To satisfy the decoupling condition of the d,q axis, “virtual impedance” is introduced inlow-voltage AC MG [25,26]. After decoupling, structure of the q axis is chosen to analyze because d,qshare the same structure. Considering the pulse-width modulation (PWM) and the sampling process,the control diagram is shown in Figure 4.


In Figure 3, Vcd and Vcq are the d,q axis voltage component of converters and Vod and Voq are the d,q axis voltage components of the filter; Lf is the inductance of filter; Kip and Kii are the proportional and integral parameters of the proportional-integral (PI) controller.

To satisfy the decoupling condition of the d,q axis, “virtual impedance” is introduced in low-voltage AC MG [25,26]. After decoupling, structure of the q axis is chosen to analyze because d,q share the same structure. Considering the pulse-width modulation (PWM) and the sampling process, the control diagram is shown in Figure 4.

_

_

Figure 4. Control structure diagram of the inner current loop.

As shown above, Ts is the time constant of sampling and its value is pretty small. 1/(Tss + 1) and 1/(0.5Tss + 1) are the delay modules of sampling and PWM, and they can approximately merge into 1/(1.5Tss + 1) [27]. Set up the switching frequency of the system as: fs = 20 kHz, Ts = 1/fs = 0.05 ms. The gain of the modulator is Kpwm = Udc/2 = 400.

Then, the open-loop transfer function is as shown

0 ( )(1.5 1)

ip pwmiq

s f

K KG s

T s L s=

+.

(1a)

Similarly, the open-loop transfer function of the d-axis is

0 ( )(1.5 1)

ip pwmid

s f

K KG s

T s L s=

+ . (1b)

In order to impose that the cut-off frequency of control is 0.1 times the switching frequency, the following relations must be satisfied:

0

210

20lg ( ) 0

sx

xiq

f

G j

πω

ω

=

=,

(2)

where ωx is the designed cut-off frequency. If the switching frequency of the PWM modulator is 20 kHz, set up the inductor-capacitor (LC)

passive filter’s inductance Lf as 1.5 mH. Kip ≈ 0.065 can be obtained. The design principles of the converter output LC passive filter is

10 / 52 1/ (2 )

n c s

c cf f

f f ff L f Cπ π

≤ ≤=

, (3)

where fc is the designed resonant frequency of filter, and fn is the fundamental frequency. Setting up Lf is 1.5 mH and Cf is 20 μF.

Based on the above analysis, the control structure of the PQ control method is shown in Figure 5 [28].

Figure 4. Control structure diagram of the inner current loop.

As shown above, Ts is the time constant of sampling and its value is pretty small. 1/(Tss + 1) and1/(0.5Tss + 1) are the delay modules of sampling and PWM, and they can approximately merge into1/(1.5Tss + 1) [27]. Set up the switching frequency of the system as: fs = 20 kHz, Ts = 1/fs = 0.05 ms.The gain of the modulator is Kpwm = Udc/2 = 400.

Then, the open-loop transfer function is as shown

Giq0(s) =KipKpwm

(1.5Tss + 1)L f s. (1a)

Similarly, the open-loop transfer function of the d-axis is

Gid0(s) =KipKpwm

(1.5Tss + 1)L f s. (1b)

In order to impose that the cut-off frequency of control is 0.1 times the switching frequency,the following relations must be satisfied:

ωx = 2π fs10

20lg∣∣Giq0(jωx)

∣∣ = 0, (2)

where ωx is the designed cut-off frequency.If the switching frequency of the PWM modulator is 20 kHz, set up the inductor-capacitor (LC)

passive filter’s inductance Lf as 1.5 mH. Kip ≈ 0.065 can be obtained. The design principles of theconverter output LC passive filter is

10 fn ≤ fc ≤ fs/52π fcL f = 1/(2π fcC f )

, (3)

where fc is the designed resonant frequency of filter, and fn is the fundamental frequency. Setting up Lfis 1.5 mH and Cf is 20 µF.

Based on the above analysis, the control structure of the PQ control method is shown inFigure 5 [28].


−uq

Q

1.5Upcc

Q

*

KPi sKPp+ Giq(s)Icq* 1

CfsIcq V0q

−ud

P

P

*

Gid(s)Icd* 1Cfs

Icd V0d KQi sKQp+

iq1

iq2

id1

id2

iq

id

−XR +X2 2

RR +X2 2

XR +X2 2

RR +X2 2

−1.5Upcc

−

−

Figure 5. Block diagram for PQ (real and reactive power) control.

As shown, Kpp and Kpi are the proportional and integral parameters of the PI controller for active power, respectively; KOp and KOi are appropriate parameters for reactive power. Due to the symmetry of the two parts, it makes KPp = KQp and KPi = KQi. In addition, P*, Q* are the reference values of active and reference power, respectively. After decoupling, the open-loop transfer function is shown

2

2

1.5 ( )( ) ( )

1.5 ( )( ) ( )

pcc Qp QiQ id

f

pcc Pp PiP iq

f

U K s KG s G s

C Xs

U K s KG s G s

C Xs

+=

+ = .

(4)

3.2. Inductive Droop Control

The DGs with droop control can independently adjust the balance of frequency and voltage and control the operation of MG as the main control micro source. Its control structure still adopts voltage-current double loop structure, where the principle of its current control loop is similar to Section 2.1. Likewise, the q-axis structure of voltage control loop after decoupling is shown in Figure 6.

_

Figure 6. Structure diagram of voltage loop control.

The open-loop transfer function in this structure is [29]

0 2

( )( )

( 1)iq vp vi

vqsf

G s K s KG s

C T s s+

=+

（）

. (5)

It is necessary to install an LC passive filter at the converter outlet to filter high order harmonics, which can be designed as Equation (3).

This paper adopts: fn = 50 Hz, fs = 20 kHz, fc = 1000 Hz. Considering the voltage drop on filter inductance, Lf = 1.5 mH, and Cf = 16.89 μF can be obtained. At this point, the PI parameters of outer loop must be satisfied:

020lg ( ) 0xvqG jω =. (6)


As shown, Kpp and Kpi are the proportional and integral parameters of the PI controller for activepower, respectively; KOp and KOi are appropriate parameters for reactive power. Due to the symmetryof the two parts, it makes KPp = KQp and KPi = KQi. In addition, P*, Q* are the reference values of activeand reference power, respectively. After decoupling, the open-loop transfer function is shown

GQ(s) =1.5Upcc(KQps+KQi)

C f Xs2 Gid(s)

GP(s) =1.5Upcc(KPps+KPi)

C f Xs2 Giq(s). (4)


The DGs with droop control can independently adjust the balance of frequency and voltageand control the operation of MG as the main control micro source. Its control structure still adoptsvoltage-current double loop structure, where the principle of its current control loop is similar toSection 2.1. Likewise, the q-axis structure of voltage control loop after decoupling is shown in Figure 6.


−uq

Q

1.5Upcc

Q

*

KPi sKPp+ Giq(s)Icq* 1

CfsIcq V0q

−ud

P

P

*

Gid(s)Icd* 1Cfs

Icd V0d KQi sKQp+

iq1

iq2

id1

id2

iq

id

−XR +X2 2

RR +X2 2

XR +X2 2

RR +X2 2

−1.5Upcc

−

−


As shown, Kpp and Kpi are the proportional and integral parameters of the PI controller for active power, respectively; KOp and KOi are appropriate parameters for reactive power. Due to the symmetry of the two parts, it makes KPp = KQp and KPi = KQi. In addition, P*, Q* are the reference values of active and reference power, respectively. After decoupling, the open-loop transfer function is shown

2

2

1.5 ( )( ) ( )

1.5 ( )( ) ( )

pcc Qp QiQ id

f

pcc Pp PiP iq

f

U K s KG s G s

C Xs

U K s KG s G s

C Xs

+=

+ = .

(4)


The DGs with droop control can independently adjust the balance of frequency and voltage and control the operation of MG as the main control micro source. Its control structure still adopts voltage-current double loop structure, where the principle of its current control loop is similar to Section 2.1. Likewise, the q-axis structure of voltage control loop after decoupling is shown in Figure 6.

_



0 2

( )( )

( 1)iq vp vi

vqsf

G s K s KG s

C T s s+

=+

（）

. (5)

It is necessary to install an LC passive filter at the converter outlet to filter high order harmonics, which can be designed as Equation (3).

This paper adopts: fn = 50 Hz, fs = 20 kHz, fc = 1000 Hz. Considering the voltage drop on filter inductance, Lf = 1.5 mH, and Cf = 16.89 μF can be obtained. At this point, the PI parameters of outer loop must be satisfied:

020lg ( ) 0xvqG jω =. (6)



Gvq0(s) =Giq(s)(Kvps + Kvi)

C f (Tss + 1)s2 . (5)

It is necessary to install an LC passive filter at the converter outlet to filter high order harmonics,which can be designed as Equation (3).

This paper adopts: fn = 50 Hz, fs = 20 kHz, fc = 1000 Hz. Considering the voltage drop on filterinductance, Lf = 1.5 mH, and Cf = 16.89 µF can be obtained. At this point, the PI parameters of outerloop must be satisfied:

20lg∣∣Gvq0(jωx)

∣∣ = 0. (6)

Taking the stability and rapidity into account, Kvp = 0.1 and Kvi = 407.65 can be obtained; then,Kvi = 400.

Energies 2018, 11, 2223 7 of 18

The converter is connected to the point of common coupling (PCC). The voltage of this point is aplanned constant and selected as the reference voltage, that is: Upcc < 0 = Upcc + j0, the output voltageof converter filter is: V0 < θ = V0d + jV0q. Supposing that the impedance of low-voltage MG is Rl + jXl,where Xl = ωLl, ω is the AC signal frequency, Ll is the equivalent inductance of transmission line; then,the voltage and current relation of the d and q axes are

V0d −Upcc = (Rl + Lls)I0d − Xl I0qV0q − 0 = (Rl + Lls)I0q + Xl I0d

. (7)

The expression of small signal quantity of the current as shown is∆I0q = ∆I0q1 + ∆I0q2 = −Ha × ∆V0d + Hb × ∆V0q∆I0d = ∆I0d1 + ∆I0d2 = Hb × ∆V0d + Ha × ∆V0q

, (8)

where Ha and Hb can be expressed as

Ha =Xl

X2l + (Rl + Lls)

2 ; Hb =(Rl + Lls)

X2l + (Rl + Lls)

2 . (9)

Actually, I0d and I0q are DC variables, and their differential results are negligible for Lls and muchsmaller than Xl.

P + jQ = Upcc(I0d − jI0q) is the power equation in the dq0 coordinate system, and the power smallsignal model can be obtained:

∆Q = −1.5Upcc(∆I0q1 + ∆I0q2)

∆P = 1.5Upcc(∆I0d1 + ∆I0d2). (10)

For double loop structure, the virtual impedance is equivalent to introducing a logical inductivefeedback link in essence, which corrects the reference voltage of dual-loop control and controls theoutput power further.

Supposing that θ is quite small, then V0q* = E*sinθ ≈ E*θ, V*

0d ≈ E*, where E* is the referencevoltage from reactive droop loop. Furthermore, it can be considered that ∆V*

0q ≈ E∆θ because changesof E* are much smaller than that of θ, where E is the reference voltage before reactive droop loop.The control structure is shown in Figure 7 [26].


Taking the stability and rapidity into account, Kvp = 0.1 and Kvi = 407.65 can be obtained; then, Kvi = 400.

The converter is connected to the point of common coupling (PCC). The voltage of this point is a planned constant and selected as the reference voltage, that is: Upcc < 0 = Upcc + j0, the output voltage of converter filter is: V0 < θ = V0d + jV0q. Supposing that the impedance of low-voltage MG is Rl + jXl, where Xl = ωLl, ω is the AC signal frequency, Ll is the equivalent inductance of transmission line; then, the voltage and current relation of the d and q axes are

0 0 0

0 0 0

( )

0 ( )d pcc l l d l q

q l l q l d

V U R L s I X I

V R L s I X I

− = + − − = + + .

(7)

The expression of small signal quantity of the current as shown is

0 0 1 0 2 0 0

0 0 1 0 2 0 0

q q q a d b q

d d d b d a q

I I I H V H V

I I I H V H V

∆ = ∆ + ∆ = − ∆ + ∆

∆ = ∆ + =

×

∆ ∆ × ∆

×

+×

,

(8)

where Ha and Hb can be expressed as

2 2 2 2

( )

( ) ( )l l l

a b

l l l l l l

X R L sH H

X R L s X R L s

+= =

+ + + +； . (9)

Actually, I0d and I0q are DC variables, and their differential results are negligible for Lls and much smaller than Xl.

P + jQ = Upcc(I0d − jI0q) is the power equation in the dq0 coordinate system, and the power small signal model can be obtained:

0 1 0 2

0 1 0 2

1.5

1.5

pcc q q

pcc d d

Q U I I

P U I I

∆ = ∆ + ∆

∆ = ∆ + ∆

−

（）

（）. (10)

For double loop structure, the virtual impedance is equivalent to introducing a logical inductive feedback link in essence, which corrects the reference voltage of dual-loop control and controls the output power further.

Supposing that θ is quite small, then V0q* = E*sinθ ≈ E*θ, V*0d ≈ E*, where E* is the reference voltage from reactive droop loop. Furthermore, it can be considered that ΔV*0q ≈ EΔθ because changes of E* are much smaller than that of θ, where E is the reference voltage before reactive droop loop. The control structure is shown in Figure 7 [26].

Figure 7. Droop control structure diagram in low-voltage MG.

After reasonable setting of virtual impedance, it is equivalent that the previous line impedance connects in series with a reactance Xv, which satisfies the following relationship Xv + Xv >> Rl, and then Rl in Ha and Hb can be neglected, that is Hb = 0, indicating the realization of decoupling in droop control.

Figure 7. Droop control structure diagram in low-voltage MG.

After reasonable setting of virtual impedance, it is equivalent that the previous line impedanceconnects in series with a reactance Xv, which satisfies the following relationship Xv + Xv >> Rl,and then Rl in Ha and Hb can be neglected, that is Hb = 0, indicating the realization of decoupling indroop control.

Energies 2018, 11, 2223 8 of 18

4. Small Signal Modeling of Three-Source MG

Theoretical analysis of a three-source MG with all DGs working at the same time is performed onthe basis of model built in Section 1.1. To help analyze the stability of system, the electrical quantitiesare expressed in steady state small signal equations [30,31], according to the relationship of voltageand current in dq0 coordinate system:

∆Vodn − ∆UGd = (Rn +Xnω s)∆idn − Xn∆iqn

∆Voqn − ∆UGq = (Rn +Xnω s)∆iqn + Xn∆idn

. (11)

Generally, the differential terms in Equation (11) has less influence and can be ignored. Therefore,the system model satisfies Equation (12):

∆Vod1 − ∆UGd = R1∆id1 − X1∆iq1

∆Voq1 − ∆UGq = R1∆iq1 + X1∆id1∆Vod2 − ∆UGd = R2∆id2 − X2∆iq2

∆Voq2 − ∆UGq = R2∆iq2 + X2∆id2∆Vod3 − ∆UGd = R3∆id3 − X3∆iq3

∆Voq3 − ∆UGq = R1∆iq3 + X1∆id30− ∆UGd = R4∆id4 − X4∆iq4

0− ∆UGq = R2∆iq4 + X4∆id4∆id1 + ∆id2 + ∆id3 + ∆id4 = 0∆iq1 + ∆iq2 + ∆iq3 + ∆iq4 = 0

. (12)

Assuming that ∆Vod1, ∆Voq1, ∆Vod2, ∆Voq2, ∆Vod3, ∆Voq3, R1, X1, R2, X2, R3, X3, R4, X4 are known,∆id1, ∆iq1, ∆id2, ∆iq2, ∆id3, ∆iq3, ∆id4, ∆iq4 can be expressed with the above quantities by solvingEquation (12), ∆UGd, ∆jUGq by Equation (13):

∆UGd = J1∆Vod1 + J2∆Vod2 + J3∆Vod3 + J4∆Voq1 + J5∆Voq2 + J6∆Voq3

∆UGq = J7∆Vod1 + J8∆Vod2 + J9∆Vod3 + J10∆Voq1 + J11∆Voq2 + J12∆Voq3. (13)

It is clear that X1 >> R1, X2 >> R2 [25,26] when DG1 and DG2 adopt the droop control method,and then the influence of line resistance can be ignored, that is: R1 = R2 = 0. Suppose that the load ismostly active and then: X4 = 0. Therefore, the coefficients in Equation (13) can be expressed as

J1 = J10 = [R23R2

4(X1X2 + X22) + R2

4X2X3(X1X2 + X1X3 + X2X3)]/DdJ2 = J11 = [R2

3R24(X1X2 + X2

1) + R24X1X3(X1X2 + X1X3 + X2X3)]/Dd

J3 = J12 = [R3R4X21 X2

2 + R24X1X2(X1X2 + X1X3 + X2X3)]/Dd

J4 = −J7 = [R4X1X22(R2

3 + R3R4 + X23)]/Dd

J5 = −J8 = [R4X21 X2(R2

3 + R3R4 + X23)]/Dd

J6 = −J9 = [−R3R24X1X2(X1 + X2) + R4X2

1 X22 X3]/Dd

Dd = R23R2

4(X1 + X2)2 + (R1 + R2)

2X21 X2

2 + X21 X2

2 X23 + 2R2

4X1X2X3(X1 + X2 + X3) + R24X2

3(X21 + X2

3)

. (14)

After ignoring the differential terms, model of droop control is shown by Equations (13) and (14)is obtained from model of PQ control:

[∆E∗1 −3n1Upcc

2X1(∆Vod1 − ∆UGd)]Gvd1(s) = ∆Vod1

[∆ω∗01 −3m1Upcc

2X1(∆Voq1 − ∆UGq)]Gvq1(s) = ∆Voq1

[∆E∗2 −3n2Upcc

2X2(∆Vod2 − ∆UGd)]Gvd2(s) = ∆Vod2

[∆ω∗02 −3m2Upcc

2X2(∆Voq2 − ∆UGq)]Gvq2(s) = ∆Voq2

, (15)

Energies 2018, 11, 2223 9 of 18

GQ(s)

∆Q∗3 + 1.5Upcc[−X3M + R3N]= ∆Vod3

GP(s)

∆P∗3 − 1.5Upcc[R3M + X3N]= ∆Vod3

M = (∆Vod3−∆UGd)

X23+R2

3

N =(∆Voq3−∆UGq)

X23+R2

3

. (16)

To simplify the equations, setting up H1 = 1.5n1Upcc/X1, H2 = 1.5m1Upcc/X1, H3 = 1.5n2Upcc/X2,H4 = 1.5m2Upcc/X2, H5 = 1.5UpccX3/(X2

3 + R23), H6 = 1.5UpccR3/(X2

3 + R23), by solving simultaneously

Equations (15) and (16), Equation (17) is obtained.

[∆E∗1 − H1(∆Vod1 − ∆UGd)]Gvd1(s) = ∆Vod1[∆ω∗01 − H2(∆Voq1 − ∆UGq)]Gvq1(s) = ∆Voq1

[∆E∗2 − H3(∆Vod2 − ∆UGd)]Gvd2(s) = ∆Vod2[∆ω∗02 − H4(∆Voq2 − ∆UGq)]Gvq2(s) = ∆Voq2

GQ(s)[∆Q∗3 − H5(∆Vod3 − ∆UGd)+

H6(∆Voq3 − ∆UGq)] = ∆Vod3GP(s)[∆P∗3 − H6(∆Vod3 − ∆UGd)+

H5(∆Voq3 − ∆UGq)] = ∆Vod3

, (17)

where ∆E∗1 and ∆ω0i* are the reference voltage and reference frequency of DGi (I = 1, 2), which adopts

droop control.By solving Equations (13) and (17), Equation (18) can be obtained:[

∆UGd ∆UGq

]T= H2×6

G

[∆E∗1 ∆ω∗01 ∆E∗2 ∆ω∗02 ∆Q∗3 ∆P∗3

]T. (18)

On the premise of guaranteeing the stability of the inner loop, Gid(s) = Giq(s) = 1 is reasonable [21,32];therefore, Gvd(s) = Gvq(s) = 1. In addition, HG is a matrix of 2 × 6, whose simplified one is shown inEquation (19): [

∆UGd∆UGq

]=

C1(s)D(s)

C2(s)D(s)

C3(s)D(s)

C4(s)D(s)

C5(s)D(s)

C6(s)D(s)

F1(s)D(s)

F2(s)D(s)

F3(s)D(s)

F4(s)D(s)

F5(s)D(s)

F6(s)D(s)

×[

∆E∗1 ∆ω∗01 ∆E∗2 ∆ω∗02 ∆Q∗3 ∆P∗3]T

.

(19)

As shown above, Equation (19) is the small-signal control model of the islanded MG, elementsof whose coefficient matrix share the same denominator named D(s). By analyzing the characteristicequation D(s) = 0, the stability of the system can be determined.

5. Simulation and Stability Analysis

5.1. Parameter Setting and Calculation

Based on Figure 1, DG1 and DG2 should satisfy: P1 = 30,000− (f − 50)/m1, P2 = 30,000− (f − 50)/m2,Q1 = 5000 − (E1 − 311)/n1, Q2 = 5000 − (E2 − 311)/n2.

Using the parameters in Appendix A, H1–H6 can be obtained: H1 = 466.5n1, H2 = 466.5m1,H3 = 466.5n2, H4 = 466.5m2, H5 = 208.6 and H6 = 417.2.

5.2. Influence of Active Droop Coefficient m1 and m2 on Stability

To explore the impact of single variable m1 on system stability, a priority assignment is firstimplemented: n1 = n2 = 0.001 V/var, KPp = KQp = 0.0005, KPi = KQi = 0.5.

Energies 2018, 11, 2223 10 of 18

(1) When m2 = 5 × 10−5 Hz/W, the characteristic equation D(s) = 0 is

(m1 + 0.00242)s5 + (234m1 + 0.542)s4 + (125m1 + 0.392)s3

+(9860m1 + 42.3)s2 + (3790m1 + 15.7)s + (58, 000m1 + 122) = 0. (20)

The root locus equation as Equation (20) whose gain is m1 can be acquired by processingEquation (21):

m1W1(s) + 1 = 0, (21)

where W1(s) and the similar expressions hereafter in this paper can be found in Appendix B.The root locus of D(s) when m1 varies from 0 to +∞ is shown in Figure 8.


As shown above, Equation (19) is the small-signal control model of the islanded MG, elements of whose coefficient matrix share the same denominator named D(s). By analyzing the characteristic equation D(s) = 0, the stability of the system can be determined.

5. Simulation and Stability Analysis

5.1. Parameter Setting and Calculation

Based on Figure 1, DG1 and DG2 should satisfy: P1 = 30,000 − (f − 50)/m1, P2 = 30,000 − (f − 50)/m2, Q1 = 5000 − (E1 − 311)/n1, Q2 = 5000 − (E2 − 311)/n2.

Using the parameters in Appendix A, H1–H6 can be obtained: H1 = 466.5n1, H2 = 466.5m1, H3 = 466.5n2, H4 = 466.5m2, H5 = 208.6 and H6 = 417.2.

5.2. Influence of Active Droop Coefficient m1 and m2 on Stability

To explore the impact of single variable m1 on system stability, a priority assignment is first implemented: n1 = n2 = 0.001 V/var, KPp = KQp = 0.0005, KPi = KQi = 0.5.

(1) When m2 = 5 × 10−5 Hz/W, the characteristic equation D(s) = 0 is

5 4 31 1 1

21 1 1 0

( + 0.00242) (234 0.542) (125 0.392)

(9860 42.3) (3790 15.7) (58,000 122)

m s m s m s

m s m s m

+ + + +

+ + + + + + = . (20)

The root locus equation as Equation (20) whose gain is m1 can be acquired by processing Equation (21):

1 1( ) + 1= 0mW s , (21)

where W1(s) and the similar expressions hereafter in this paper can be found in Appendix B. The root locus of D(s) when m1 varies from 0 to +∞ is shown in Figure 8.

Figure 8. The root locus of D(s) following m1 changes when m2 = 5 × 10−5 Hz/W.

There are five branches, S1 starts from (−233.8x, 0) to (−225.3, 0) and always stays in left half-plane and far away from the imaginary axis, so it is not shown in the picture; S2 and S3 locate in left half-plane and have no impact on stability; S4 and S5 move towards the positive direction with the change of m1, leading to the decline in stability, they finally cross the imaginary axis (m1 = 3.2 × 10−5) and enter the right half-plane indicating that the system is no longer stable. Therefore, the proper range of m1 to ensure system stability is [0, 3.2 × 10−5) in this condition.

Theoretically, the smaller m1 is, the weaker the impact of change in DG1’s output power on system’s frequency is, especially when m1 = 0, DG1 works in constant frequency control mode and has the strongest stability. At this point, DG1 has theoretically infinite power supply, which owns the absolute frequency stability and peak regulating ability. However, this assumption is not


There are five branches, S1 starts from (−233.8x, 0) to (−225.3, 0) and always stays in left half-planeand far away from the imaginary axis, so it is not shown in the picture; S2 and S3 locate in left half-planeand have no impact on stability; S4 and S5 move towards the positive direction with the change of m1,leading to the decline in stability, they finally cross the imaginary axis (m1 = 3.2 × 10−5) and enter theright half-plane indicating that the system is no longer stable. Therefore, the proper range of m1 toensure system stability is [0, 3.2 × 10−5) in this condition.

Theoretically, the smaller m1 is, the weaker the impact of change in DG1’s output power onsystem’s frequency is, especially when m1 = 0, DG1 works in constant frequency control mode andhas the strongest stability. At this point, DG1 has theoretically infinite power supply, which owns theabsolute frequency stability and peak regulating ability. However, this assumption is not consistentwith the energy attributes of MG. That is, MG’s output power is limited and relies on other MGs’cooperation, so m1 cannot be too small. On the other hand, a large value of m1 also results in the fastoscillation of system frequency under small power fluctuation, which is harmful to system stabilityand this situation is shown in Figure 8 when m1 > 3.2 × 10−5.

When m1 = 2.5 × 10−5 Hz/W and m2 = 5 × 10−5 Hz/W, the bus voltage waveform is shown inFigure 9. Because 2.5 × 10−5 ∈ [0, 3.2 × 10−5) and the bus voltage is stable, it means that the m1 in thisrange ensures microgrid stable operation, which can verify the correctness of above analysis.

Energies 2018, 11, 2223 11 of 18


consistent with the energy attributes of MG. That is, MG’s output power is limited and relies on other MGs’ cooperation, so m1 cannot be too small. On the other hand, a large value of m1 also results in the fast oscillation of system frequency under small power fluctuation, which is harmful to system stability and this situation is shown in Figure 8 when m1 > 3.2 × 10−5.

When m1 = 2.5 × 10−5 Hz/W and m2 = 5 × 10−5 Hz/W, the bus voltage waveform is shown in Figure 9. Because 2.5 × 10−5 ∈ [0, 3.2 × 10−5) and the bus voltage is stable, it means that the m1 in this range ensures microgrid stable operation, which can verify the correctness of above analysis.

Time/s

Busb

ar v

olta

ge/V

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−800

−600

−400

−200

0

200

400

600

800

Figure 9. The bus voltage waveform when m1 = 2.5 × 10−5 Hz/W and m2 = 5 × 10−5 Hz/W.

(2) When m2 = 10 × 10−5 Hz/W, the root locus equation whose gain is m1 is

1 2 ( ) + 1= 0mW s . (22)

The root locus of D(s) when m1 varies from 0 to +∞ is shown in Figure 10.


The locus is so similar to Figure 8 that it won’t be described again. The proper range of m1 is [0, 2.9 × 10−5), and system stability declines with the increase of m1.

When m1 = 2 × 10−5 Hz/W and m2 = 10 × 10−5 Hz/W, the bus voltage waveform is shown in Figure 11. The waveform is similar to Figure 9, without more detailed analysis.



m1W2(s) + 1 = 0. (22)



consistent with the energy attributes of MG. That is, MG’s output power is limited and relies on other MGs’ cooperation, so m1 cannot be too small. On the other hand, a large value of m1 also results in the fast oscillation of system frequency under small power fluctuation, which is harmful to system stability and this situation is shown in Figure 8 when m1 > 3.2 × 10−5.

When m1 = 2.5 × 10−5 Hz/W and m2 = 5 × 10−5 Hz/W, the bus voltage waveform is shown in Figure 9. Because 2.5 × 10−5 ∈ [0, 3.2 × 10−5) and the bus voltage is stable, it means that the m1 in this range ensures microgrid stable operation, which can verify the correctness of above analysis.

Time/s

Busb

ar v

olta

ge/V

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−800

−600

−400

−200

0

200

400

600

800



1 2 ( ) + 1= 0mW s . (22)




When m1 = 2 × 10−5 Hz/W and m2 = 10 × 10−5 Hz/W, the bus voltage waveform is shown in Figure 11. The waveform is similar to Figure 9, without more detailed analysis.


The locus is so similar to Figure 8 that it won’t be described again. The proper range of m1 is[0, 2.9 × 10−5), and system stability declines with the increase of m1.

When m1 = 2 × 10−5 Hz/W and m2 = 10 × 10−5 Hz/W, the bus voltage waveform is shown inFigure 11. The waveform is similar to Figure 9, without more detailed analysis.Energies 2018, 11, x FOR PEER REVIEW 12 of 19

Time/s

Busb

ar v

olta

ge/V

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−800

−600

−400

−200

0

200

400

600

800

Figure 11. The bus voltage waveform when m1 = 2 × 10−5 Hz/W and m2 = 10 × 10−5 Hz/W.

(3) When m2 = 2.5 × 10−5 Hz/W, the root locus equation whose gain is m1 is

1 3( ) + 1= 0mW s . (23)


Figure 12. The root locus of D(s) following m1 changes when m2 = 2.5 × 10−5 Hz/W.


When m1 = 5 × 10−5 Hz/W and m2 = 2.5 × 10−5 Hz/W, the bus voltage waveform is shown in Figure 13. The waveform is similar to Figure 9, without more detailed analysis.

Time/s

Busb

ar v

olta

ge/V

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−800

−600

−400

−200

0

200

400

600

800

Figure 13. The bus voltage waveform when m1 = 5 × 10−5 Hz/W and m2 = 2.5 × 10−5 Hz/W.



Energies 2018, 11, 2223 12 of 18

m1W3(s) + 1 = 0. (23)



Time/s

Busb

ar v

olta

ge/V

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−800

−600

−400

−200

0

200

400

600

800



1 3( ) + 1= 0mW s . (23)





Time/s

Busb

ar v

olta

ge/V

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−800

−600

−400

−200

0

200

400

600

800

Figure 13. The bus voltage waveform when m1 = 5 × 10−5 Hz/W and m2 = 2.5 × 10−5 Hz/W.


The locus is so similar to Figure 8 that it won’t be described again. The proper range of m1 is[0, 7.21 × 10−5), and system stability declines with the increase of m1.

When m1 = 5 × 10−5 Hz/W and m2 = 2.5 × 10−5 Hz/W, the bus voltage waveform is shown inFigure 13. The waveform is similar to Figure 9, without more detailed analysis.


Time/s

Busb

ar v

olta

ge/V

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−800

−600

−400

−200

0

200

400

600

800



1 3( ) + 1= 0mW s . (23)





Time/s

Busb

ar v

olta

ge/V

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−800

−600

−400

−200

0

200

400

600

800

Figure 13. The bus voltage waveform when m1 = 5 × 10−5 Hz/W and m2 = 2.5 × 10−5 Hz/W. Figure 13. The bus voltage waveform when m1 = 5 × 10−5 Hz/W and m2 = 2.5 × 10−5 Hz/W.

In summary, the comparison of range for m1 when m2 is different is shown in Table 1. In Table 1, m1

is less than 2.9 × 10−5 when m2 is 10 × 10−5. It is the reason why bus voltage is unstable in Section 1.2.The allowable range of m1 narrows with the increase of m2. From the perspective of system stability,the active droop coefficients of two DGs are supposed to be coordinated and restricted in a lower rangeso that the unit power loss won’t cause large change in frequency and system frequency collapse.

Table 1. The influence of active droop coefficients between DG1 and DG2.

m2 (Hz/W) Range of m1 (Hz/W)

2.5 × 10−5 [0, 7.21 × 10−5)5 × 10−5 [0, 3.2 × 10−5)

10 × 10−5 [0, 2.9 × 10−5)

5.3. Influence of Reactive Droop Coefficient n1 and n2 on Stability

To explore the impact of single variable n1 on system stability, a priority assignment is firstimplemented: m1 = m2 = 2.5 × 10−5 Hz/W, KPp = KQp = 0.0005, KPi = KQi = 0.5.

Energies 2018, 11, 2223 13 of 18

(1) When n2 = 0.002 V/var, the root locus of the equation D(s) = 0 whose gain is n1 is

n1W4(s) + 1 = 0. (24)

The root locus when n1 varies from 0 to +∞ is shown in Figure 14.


In summary, the comparison of range for m1 when m2 is different is shown in Table 1. In Table 1, m1 is less than 2.9 × 10−5 when m2 is 10 × 10−5. It is the reason why bus voltage is unstable in Section 1.2. The allowable range of m1 narrows with the increase of m2. From the perspective of system stability, the active droop coefficients of two DGs are supposed to be coordinated and restricted in a lower range so that the unit power loss won’t cause large change in frequency and system frequency collapse.

Table 1. The influence of active droop coefficients between DG1 and DG2.

m2 (Hz/W) Range of m1 (Hz/W) 2.5 × 10−5 [0, 7.21 × 10−5) 5 × 10−5 [0, 3.2 × 10−5)

10 × 10−5 [0, 2.9 × 10−5)

5.3. Influence of Reactive Droop Coefficient n1 and n2 on Stability

To explore the impact of single variable n1 on system stability, a priority assignment is first implemented: m1 = m2 = 2.5 × 10−5 Hz/W, KPp = KQp = 0.0005, KPi = KQi = 0.5.

(1) When n2 = 0.002 V/var, the root locus of the equation D(s) = 0 whose gain is n1 is

1 4 ( ) + 1 0nW s = . (24)

The root locus when n1 varies from 0 to +∞ is shown in Figure 14.

Figure 14. The root locus of D(s) following n1 changes when n2 = 0.002 V/var.

There are six branches, S1 first moves towards left and then turns right and always stays in the left half-plane, indicating that the system stability first enhances and then degrades; S2 and S3 are located in the left half-plane and have no impact on stability; S4 and S5 move towards the positive direction with the change of n1, leading to the decline in stability, they finally cross the imaginary axis (n1 = 0.00147) and enter the right half-plane indicating that the system is no longer stable. Therefore, the proper range of n1 to ensure system stability is [0, 0.00147) in this condition.

Theoretically, the smaller n1 is, the weaker the impact of change in DG1’s output reactive power on system’s voltage is. Especially when n1 = 0, DG1 works in constant voltage control mode and has the strongest stability. At this point, DG1 has theoretically infinite reactive power supply, which owns the absolute voltage stability and reactive power compensation ability. However, this assumption is not consistent with the energy attributes of MG. That is, MG’s output reactive power is limited and relies on other MGs’ cooperation, so n1 cannot be too small. On the other hand, a large value of n1 also results in voltage collapse under small reactive power fluctuation, and this instable situation is shown in picture when n1 > 0.00147.


There are six branches, S1 first moves towards left and then turns right and always stays in theleft half-plane, indicating that the system stability first enhances and then degrades; S2 and S3 arelocated in the left half-plane and have no impact on stability; S4 and S5 move towards the positivedirection with the change of n1, leading to the decline in stability, they finally cross the imaginary axis(n1 = 0.00147) and enter the right half-plane indicating that the system is no longer stable. Therefore,the proper range of n1 to ensure system stability is [0, 0.00147) in this condition.

Theoretically, the smaller n1 is, the weaker the impact of change in DG1’s output reactive poweron system’s voltage is. Especially when n1 = 0, DG1 works in constant voltage control mode and hasthe strongest stability. At this point, DG1 has theoretically infinite reactive power supply, which ownsthe absolute voltage stability and reactive power compensation ability. However, this assumption isnot consistent with the energy attributes of MG. That is, MG’s output reactive power is limited andrelies on other MGs’ cooperation, so n1 cannot be too small. On the other hand, a large value of n1 alsoresults in voltage collapse under small reactive power fluctuation, and this instable situation is shownin picture when n1 > 0.00147.

When n1 = 0.001 V/var and n2 = 0.002 V/var, the bus voltage waveform is shown in Figure 15.Because 0.001 ∈ [0, 0.00147) and the bus voltage is stable, it means that the n1 in this range ensuresmicrogrid stable operation, which can verify the correctness of the above analysis.


When n1 = 0.001 V/var and n2 = 0.002 V/var, the bus voltage waveform is shown in Figure 15. Because 0.001 ∈ [0, 0.00147) and the bus voltage is stable, it means that the n1 in this range ensures microgrid stable operation, which can verify the correctness of the above analysis.

Time/s

Busb

ar v

olta

ge/V

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−800

−600

−400

−200

0

200

400

600

800

Figure 15. The bus voltage waveform when n1 = 0.001 V/var and n2 = 0.002 V/var.

(2) When n2 = 0.001 V/var, the root locus equation is

1 5 ( ) + 1 0nW s = . (25)

The root locus of D(s) when n1 varies from 0 to +∞ is shown in Figure 16.


Because this locus is similar to Figure 14, it won’t be described again. The proper range of n1 is [0, 0.00169), and system stability declines with the increase of n1.

When n1 = 0.001 V/var and n2 = 0.001 V/var, the bus voltage waveform is shown in Figure 17. The waveform is similar to Figure 15, without more detailed analysis.


Energies 2018, 11, 2223 14 of 18


n1W5(s) + 1 = 0. (25)



When n1 = 0.001 V/var and n2 = 0.002 V/var, the bus voltage waveform is shown in Figure 15. Because 0.001 ∈ [0, 0.00147) and the bus voltage is stable, it means that the n1 in this range ensures microgrid stable operation, which can verify the correctness of the above analysis.

Time/s

Busb

ar v

olta

ge/V

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−800

−600

−400

−200

0

200

400

600

800



1 5 ( ) + 1 0nW s = . (25)






Because this locus is similar to Figure 14, it won’t be described again. The proper range of n1 is[0, 0.00169), and system stability declines with the increase of n1.

When n1 = 0.001 V/var and n2 = 0.001 V/var, the bus voltage waveform is shown in Figure 17.The waveform is similar to Figure 15, without more detailed analysis.Energies 2018, 11, x FOR PEER REVIEW 15 of 19

Time/s

Busb

ar v

olta

ge/V

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−800

−600

−400

−200

0

200

400

600

800



1 6 ( ) + 1 0nW s = . (26)





Time/s

Busb

ar v

olta

ge/V

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−800

−600

−400

−200

0

200

400

600

800




n1W6(s) + 1 = 0. (26)


Energies 2018, 11, 2223 15 of 18


Time/s

Busb

ar v

olta

ge/V

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−800

−600

−400

−200

0

200

400

600

800



1 6 ( ) + 1 0nW s = . (26)





Time/s

Busb

ar v

olta

ge/V

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−800

−600

−400

−200

0

200

400

600

800



Because this locus is similar to Figure 14, it won’t be described again. The proper range of n1 is[0, 0.00133), and system stability declines with the increase of n1.

When n1 = 0.001 V/var and n2 = 0.004 V/var, the bus voltage waveform is shown in Figure 19.The waveform is similar to Figure 15, without more detailed analysis.


Time/s

Busb

ar v

olta

ge/V

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−800

−600

−400

−200

0

200

400

600

800



1 6 ( ) + 1 0nW s = . (26)





Time/s

Busb

ar v

olta

ge/V

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−800

−600

−400

−200

0

200

400

600

800

Figure 19. The bus voltage waveform when n1 = 0.001 V/var and n2 = 0.004 V/var. Figure 19. The bus voltage waveform when n1 = 0.001 V/var and n2 = 0.004 V/var.

In summary, the comparison of range of n1 when n2 is different is shown in Table 2. The allowablerange of n1 narrows with the increase of n2. From the perspective of system stability, the reactivedroop coefficients of two DGs are supposed to be coordinated because it is restricted in such a lowvalue range that the unit reactive power loss won’t cause large change in voltage—thus avoiding thevicious cycle: “change in voltage-reactive power loss-enlargement change in voltage—larger reactivepower loss”.

Table 2. The influence of reactive droop coefficients between DG1 and DG2.

n2 (V/var) Range of n1 (V/var)

0.001 [0, 0.00169)0.002 [0, 0.00147)0.004 [0, 0.00133)

6. Conclusions

In this paper, research on influence of the wide application of electronic devices on system stabilityis done by a three-source MG with a multiple master-slave control method.

Energies 2018, 11, 2223 16 of 18

Different from the traditional master–slave control model, this paper established a model whereseveral master controllers coordinate and control the system in islanded mode. Then, the “virtualimpedance” was introduced to satisfy the decoupling condition and the small-signal decoupling controlmodel, which contains two main control sources and subordinate one, was also developed on thebasis of theory. Finally, the interaction between various DGs with droop control from the perspectiveof control parameters was explored with the aid of the root locus method. The results show that thevalue range of the droop coefficient for the two main DGs has mutual influence and the coordinatedallocation of parameters also affects the stability of microgrid operation. This research could serve as areference for parameter setting and contributed to study on stability of multi-source MG.

Author Contributions: All of the authors contributed to this work. H.L. and P.L. provided important commentson the modeling and designed the experiments; Y.D. and C.Z. performed the experiments and analyzed the data;Y.D. and Y.H. wrote the whole paper.

Funding: This research was funded by the National Natural Science Foundation of China (No. 51577068) and theNational High Technology Research and Development Program of China (No. 2015AA050101). And the APC wasfunded by North China Electric Power University.

Acknowledgments: The authors gratefully acknowledge the support from the National Natural ScienceFoundation of China (No. 51577068) and the National High Technology Research and Development Program ofChina (No. 2015AA050101).

Conflicts of Interest: The authors declare no conflict of interest.

Appendix A

Table A1. System basic parameter value.

Meaning of Parameter Symbol of Parameter Value of Parameter

Voltage Level/V E* 311Reference value of active power/kW P3

* 10Reference value of reactive power/kVar Q3

* −10System switching frequency/kHz fs 20

System sampling period/ms Ts 0.05Filter inductors/mH Lf1 = Lf2 = Lf3 1.5Filter capacitors/µF Cf 1 = Cf 2 = Cf 3 20

Line resistance of DG1 and DG2/Ω R1, R2 0Line resistance of DG3/Ω R3 0.2

Load resistance/Ω R4 3Line inductance of DG1 and DG2/Ω X1 = X2 1.0

Line inductance of DG3/Ω X3 0.1Load inductance/Ω X4 0

Proportional coefficient of voltage loop Kvp 0.1Integral coefficient of voltage loop Kvi 400

Proportional coefficient of current loop Kip 0.065

Appendix B

The expressions of coefficients W1(s)–W6(s) in Equations (21)–(26) are shown in (A1)–(A6).

W1(s) =412.78s5 + 96, 596s4 + 51, 663s3 + 4, 069, 653s2 + 1, 562, 487s + 23, 925, 000

s5 + 223.78s4 + 161.64s3 + 17, 462s2 + 6492s + 50, 574, (A1)

W2(s) =413.2s5 + 96, 818s4 + 51, 322s3 + 3, 997, 107s2 + 1, 536, 776s + 24, 028, 925

s5 + 224.3s4 + 161.2s3 + 17, 340s2 + 6450s + 50, 785, (A2)

W3(s) =412.7s5 + 96, 558s4 + 51, 863s3 + 4, 110, 238s2 + 1, 576, 956s + 23, 890, 375

s5 + 223.67s4 + 162s3 + 17, 538s2 + 6519s + 50, 501, (A3)

Energies 2018, 11, 2223 17 of 18

W4(s) =128, 449s5 + 32, 460, 140s4 + 17, 352, 828s3 + 2, 532, 328, 040s2 + 490, 885, 015s + 7, 337, 660, 287

s6 + 257.04s5 + 173.83s4 + 29, 655.7s3 + 8295s2 + 952, 419s, (A4)

W5(s) =128, 449s5 + 16, 237, 480s4 + 16, 087, 143s3 + 1, 266, 640, 229s2 + 483, 547, 355s + 3, 668, 830, 143

s6 + 128.6s5 + 159s4 + 14, 834s3 + 7342s2 + 476, 209s, (A5)

W6(s) =128, 449s5 + 64, 905, 460s4 + 19, 884, 196s3 + 5, 063, 703, 661s2 + 505, 560, 336s + 14, 675, 320, 574

s6 + 513.9s5 + 203.5s4 + 59, 299s3 + 10, 200s2 + 1, 904, 839s. (A6)

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